#87499

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#585 ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂³×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴
Fitting	#585	D₃
hypercenter	#41	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ #41

723 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,19,25,4,26,50,8,22,28,5,6,52)(2,51,37,21,27,60,13,35,38,10,11,59)(3,47,42,18,43,62,17,39,45,14,15,61)(7,23,29,20)(9,53,57,30,44,63,34,55,58,31,32,64)(12,36,24,49)(16,40,46,48)(33,56,41,54);(1,4,8,5)(2,10,13,21)(3,14,17,18)(6,19,26,22)(7,20,29,23)(9,30,34,31)(11,35,27,51)(12,36,24,49)(15,39,43,47)(16,40,46,48)(25,50,28,52)(32,53,44,55)(33,54,41,56)(37,59,38,60)(42,61,45,62)(57,63,58,64);(1,3,8,17)(2,9,13,34)(4,18,5,14)(6,15,26,43)(7,16,29,46)(10,31,21,30)(11,32,27,44)(12,33,24,41)(19,47,22,39)(20,48,23,40)(25,42,28,45)(35,55,51,53)(36,56,49,54)(37,57,38,58)(50,62,52,61)(59,64,60,63);(1,2)(3,9)(4,10)(5,21)(6,24)(7,27)(8,13)(11,29)(12,26)(14,30)(15,41)(16,44)(17,34)(18,31)(19,49)(20,51)(22,36)(23,35)(25,38)(28,37)(32,46)(33,43)(39,56)(40,55)(42,58)(45,57)(47,54)(48,53)(50,60)(52,59)(61,64)(62,63)

Polycyclic presentation:

2^4,3,2^3|b^2=h,c^2=h,d^2=h,f^2=h,g^2=h,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11713197, #11713199, #11713429, …
maximal normal subgroup of	#11713197, #11713199, #11713429, …
maximal quotient of	#11714609, #11714644, #11714646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20075);